Let $M$ be a smooth manifold with boundary and $p\in \partial M$. We say a tangent vector $v\in T_pM$ is inward-pointing if in a chart $x$ with $v=v^i\partial/\partial x^i$ (using the summation convention) one has $v^1>0$.
Why is this well-defined?
I.e. if $y$ is another chart around $p$ with $v=w^j \partial/\partial y^j$, how can we conclude that $w^1>0$?
I tried looking at the change of coordinates formula which says that $$w^1=\frac{\partial (y^1\circ x^{-1})}{\partial x^i}(x(p))v^i$$ but why should that be positive when $v^1>0$?
What am I missing?
Best Answer
$\newcommand{\dd}{\partial}$Near a boundary point, $M$ is modeled by the closed half-space $H = \{(x^{1}, \dots, x^{n} : x^{1} \geq 0\}$. Geometrically, a diffeomorphism representing a change of coordinates near a boundary point $p$ must preserve $H$, and so preserves the notion of a vector pointing toward the interior.
Analytically, if $v = v^{i} \dd_{i}$ is an inward-pointing tangent vector at some boundary point $p$, then the curve $\gamma(t) = p + tv$ lies in the open half-space $H^{0}$ for $t > 0$. If $$ y^{j} = \phi^{j}(x) = \phi^{j}(x^{1}, x^{2}, \dots, x^{n}) $$ is a change of coordinates at $p$, then $y^{1} \geq 0$ and $\phi \circ \gamma$ is a curve in $H$ whose initial velocity $$ w = (\phi \circ \gamma)'(0) = D\phi(p)\, v = \frac{\partial \phi^{j}}{\partial x^{i}}(p)\, v^{i}\, \frac{\dd}{\dd y^{j}} $$ points toward $H^{0}$, i.e., $$ w^{1} = \frac{\partial \phi^{1}}{\partial x^{i}}(p)\, v^{i} > 0. $$